The number of ways in which 3 identical balls | vedclass

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(B) The problem is equivalent to finding the number of non-negative integer solutions to the equation $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 = 3$,wh

Vedclass · Accepted Answer

$84$

Vedclass · Answer

(B) The problem is equivalent to finding the number of non-negative integer solutions to the equation $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 = 3$,where $x_i \ge 0$.Using the stars and bars formula,the number of ways is given by $\binom{n+r-1}{r-1}$,where $n = 3$ (identical balls) and $r = 7$ (distinct bins).Number of ways = $\binom{7+3-1}{7-1} = \binom{9}{6}$.Since $\binom{9}{6} = \binom{9}{3} = \frac{9 	imes 8 	imes 7}{3 	imes 2 	imes 1} = 3 	imes 4 	imes 7 = 84$.

The number of ways in which $3$ identical balls can be distributed into $7$ distinct bins is

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